FORCED OSCILLATIONS OF NONLINEAR HYPERBOLIC EQUATIONS WITH FUNCTIONAL ARGUMENTS

In this paper, sufficient conditions for the forced oscillations of hyperbolic equations with functional arguments of the form \[\frac{\partial^2}{\partial t^2}=a(t)\Delta u(x, t)+\sum_{i=1}^m a_i(t)\Delta u(x, \rho(t))-\sum_{j=1}^k q_j(x, t)f_j(u(x, \sigma_j(t)))+f(x, t)  \]$(x, t)\in\Omega\times[0, \infty)$ are obtained, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with piecewise smooth boundary $\partial\Omega$ and $\Delta$ is the Laplacian in Euclidean $n$-space $\mathbb{R}^n$.